Geodesic
Completed representation ring spectra of nilpotent groups
Lawson, Tyler1
1Department of Mathematics, Massachusetts Institute of Technology, Cambridge MA 02139, USA
Algebraic and Geometric Topology, Tome 6 (2006) no. 1, pp. 253-285
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In this paper, we examine the “derived completion” of the representation ring of a pro-p group Gp∧ with respect to an augmentation ideal. This completion is no longer a ring: it is a spectrum with the structure of a module spectrum over the Eilenberg–MacLane spectrum ℍℤ, and can have higher homotopy information. In order to explain the origin of some of these higher homotopy classes, we define a deformation representation ring functor R[−] from groups to ring spectra, and show that the map R[Gp∧] → R[G] becomes an equivalence after completion when G is finitely generated nilpotent. As an application, we compute the derived completion of the representation ring of the simplest nontrivial case, the p–adic Heisenberg group.

DOI : 10.2140/agt.2006.6.253
Keywords: S-algebra, R-module, completion, Bousfield localization, representation ring

Lawson, Tyler  1

1 Department of Mathematics, Massachusetts Institute of Technology, Cambridge MA 02139, USA 
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Lawson, Tyler. Completed representation ring spectra of nilpotent groups. Algebraic and Geometric Topology, Tome 6 (2006) no. 1, pp. 253-285. doi: 10.2140/agt.2006.6.253

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